Hybrid iterative scheme for a generalized equilibrium problems, variational inequality problems and fixed point problem of a finite family of κ _i -strictly pseudocontractive mappings

Abstract

In this article, by using the S-mapping and hybrid method we prove a strong convergence theorem for finding a common element of the set of fixed point problems of a finite family of κ _i -strictly pseudocontractive mappings and the set of generalized equilibrium defined by Ceng et al., which is a solution of two sets of variational inequality problems. Moreover, by using our main result we have a strong convergence theorem for finding a common element of the set of fixed point problems of a finite family of κ _i -strictly pseudocontractive mappings and the set of solution of a finite family of generalized equilibrium defined by Ceng et al., which is a solution of a finite family of variational inequality problems.

1 Introduction

Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. A mapping T of H into itself is called nonexpansive if ∥Tx - Ty∥ ≤ ∥x - y∥ for all x, y ∈ H. We denote by F(T) the set of fixed points of T (i.e., F(T) = {x ∈ H : Tx = x}) Goebel and Kirk [1] showed that F(T) is always closed convex, and also nonempty provided T has a bounded trajectory.

Recall the mapping T is said to be κ-strict pseudo-contration if there exist κ ∈ [0, 1) such that

$${∥Tx-Ty∥}^2\le {∥x-y∥}^2+\kappa {∥\left(I-T\right)x-\left(I-T\right)y∥}^2\forall x,y\in D\left(T\right).$$

(1.1)

Note that the class of κ-strict pseudo-contractions strictly includes the class of nonexpansive mappings, that is T is nonexpansive if and only if T is 0-strict pseudo-contractive. If κ = 1, T is said to be pseudo-contraction mapping. T is strong pseudo-contraction if there exists a positive constant λ ∈ (0, 1) such that T + λI is pseudo-contraction. In a real Hilbert space H (1.1) is equivalent to

$$⟨Tx-Ty,x-y⟩\le {∥x-y∥}^2-\frac{1-\kappa }{2}{∥\left(I-T\right)x-\left(I-T\right)y∥}^2\forall x,y\in D\left(T\right).$$

(1.2)

T is pseudo-contraction if and only if

$$⟨Tx-Ty,x-y⟩\le {∥x-y∥}^2\forall x,y\in D\left(T\right).$$

T is strong pseudo-contraction if there exists a positive constant λ ∈ (0, 1)

$$⟨Tx-Ty,x-y⟩\le \left(1-\lambda \right){∥x-y∥}^2\forall x,y\in D\left(T\right)$$

The class of κ-strict pseudo-contractions falls into the one between classes of nonexpansive mappings and pseudo-contraction mappings and class of strong pseudo-contraction mappings is independent of the class of κ-strict pseudo-contraction.

A mapping A of C into H is called inverse-strongly monotone, see [2] if there exists a positive real number α such that

$$⟨x-y,Ax-Ay⟩\ge \alpha {∥Ax-Ay∥}^2$$

for all x, y ∈ C.

The equilibrium problem for G is to determine its equilibrium points, i.e., the set

$$EP\left(G\right)=\left\{x\in G:G\left(x,y\right)\ge 0,\forall y\in C\right\}.$$

(1.3)

Given a mapping T : C → H, let G(x, y) = 〈Tx, y - x〉 for all x, y ∈ C. Then, z ∈ EP(F) if and only if 〈Tz, y - z〉 ≥ 0 for all y ∈ C, i.e., z is a solution of the variational inequality. Let A : C → H be a nonlinear mapping. The variational inequality problem is to find a u ∈ C such that

$$⟨v-u,Au⟩\ge 0$$

(1.4)

for all v ∈ C. The set of solutions of the variational inequality is denoted by VI(C, A).

In 2005, Combettes and Hirstoaga [3] introduced some iterative schemes of finding the best approximation to the initial data when EP(G) is nonempty and proved strong convergence theorem.

Also in [3] Combettes and Hiratoaga, following [4] define S _r : H → C by

$$S_r\left(x\right)=\left\{z\in C:G\left(z,y\right)+\frac{1}{r}⟨y-z,z-x⟩\ge 0\forall y\in C\right\}.$$

(1.5)

hey proved that under suitable hypotheses G, S _r is single-valued and firmly nonexpansive with F(S _r ) = EP(G).

Numerous problems in physics, optimization, and economics reduce to find a element of EP(G) (see, e.g., [5–16])

Let CB(H) be the family of all nonempty closed bounded subsets of H and $\mathscr{H}\left(.,.\right)$ be the Hausdorff metric on CB(H) defined as

$$\mathscr{H}\left(U,V\right)=\text{max}\left\{\underset{u\in U}{\text{sup}}d\left(u,V\right),\underset{v\in V}{\text{sup}}d\left(U,v\right)\right\},\forall U,V\in CB\left(H\right),$$

where d(u, V) = inf_v∈Vd(u, v), d(U, v) = inf_u∈Ud(u, v), and d(u, v) = ∥u - v∥.

Let C be a nonempty closed convex subset of H. Let φ : C → ℝ be a real-valued function, T : C → CB(H) a multivalued mapping and Φ : H × C × C → ℝ an equilibrium-like function, that is, Φ(w, u, v) + Φ(w, v, u) = 0 for all (w, u, v) ∈ H × C × C which satisfies the following conditions with respect to the multivalued map T : C → CB(H).

(H 1) For each fixed v ∈ C, (ω, u) ↦ Φ(ω, u, v) is an upper semicontinuous function from H × C to ℝ, that is, for (ω, u) ∈ H × C, whenever ω _n → ω and u _n → u as n → ∞,

$$\underset{n\to \infty }{\text{lim sup}}\Phi \left({\omega }_n,u_n,v\right)\le \Phi \left(\omega ,u,v\right);$$

(H 2) For each fixed (w, v) ∈ H × C, u ↦ Φ(w, u, v) is a concave function;

(H 3) For each fixed (w, u) ∈ H × C, v ↦ Φ(w, u, v) is a convex function.

In 2009, Ceng et al. [17] introduced the following generalized equilibrium problem (GEP) as follows:

$$\left(\text{GEP}\right)\left\{\begin{array}{c}\text{Find}u\in C\text{and}w\in T\left(u\right)\text{such}\text{that}\hfill \\ \Phi \left(w,u,v\right)+\phi \left(v\right)-\phi \left(u\right)\ge 0,\forall v\in C.\hfill \end{array}\right.$$

(1.6)

The set of such solutions u ∈ C of (GEP) is denote by (GEP) _s (Φ, φ).

In the case of φ ≡ 0 and Φ(w, u, v) ≡ G(u, v), then (GEP) _s (Φ, φ) is denoted by EP(G). By using Nadler's theorem they introduced the following algorithm:

Let x₁ ∈ C and w₁ ∈ T(x₁), there exists sequences {w _n } ⊆ H and {x _n }, {u _n } ⊆ C such that

$$\left\{\begin{array}{c}{w}_n\in T\left(x_n\right),∥w_n-w_{n+1}∥\le \left(1+\frac{1}{n}\right)\mathscr{H}\left(T\left(x_n\right),T\left(x_{n+1}\right)\right),\hfill \\ \Phi \left(w_n,u_n,v\right)+\phi \left(v\right)-\phi \left(u_n\right)+\frac{1}{r_n}⟨u_n-x_n,v-u_n⟩\ge 0,\forall u\in C,\hfill \\ x_{n+1}={\alpha }_{n}f\left(x_n\right)+\left(1-{\alpha }_n\right)S{u}_n,n=1,2,....\hfill \end{array}\right.$$

(1.7)

They proved a strong convergence theorem of the sequence {x _n } generated by (1.7) as follows:

Theorem 1.1. (See [17] ) Let C be a nonempty, bounded, closed, and convex subset of a real Hilbert space H and let φ : C → ℝ be a lower semicontinuous and convex functional. Let T : C → CB(H) be $\mathscr{H}$-Lipschitz continuous with constant μ, Φ : H × C × C → ℝ be an equilibrium-like function satisfying (H1)-(H3) and S be a nonexpansive mapping of C into itself such that $F\left(S\right)\cap {\left(GEP\right)}_s\left(\Phi ,\phi \right)\ne \varnothing$. Let f be a contraction of C into itself and let {x _n }, {w _n }, and {u _n } be sequences generated by (1.7), where {α _n } ⊆ [0,1] and {r _n } ⊆ (0, ∞) satisfy

$$\begin{array}{c}{\text{lim}}_{n\to \infty }{\alpha }_n=0,{\sum }_{n=1}^{\infty }{\alpha }_n=\infty ,{\sum }_{n=1}^{\infty }\left|{\alpha }_{n+1}-{\alpha }_n\right|<\infty ,\\ \text{lim }{\text{inf}}_{n\to \infty }{r}_n>0and\sum _{n=1}^{\infty }\left|r_{n+1}-r_n\right|<\infty .\end{array}$$

If there exists a constant λ > 0 such that

$$\Phi \left(w_1,T_{r_1}\left(x_1\right),T_{r_2}\left(x_2\right)\right)+\Phi \left(w_2,T_{r_2}\left(x_2\right),T_{r_1}\left(x_1\right)\right)\le -\lambda {∥T_{r_1}\left(x_1\right)-T_{r_2}\left(x_2\right)∥}^2$$

(1.8)

for all (r₁, r₂) ∈ Ξ × Ξ,(x₁, x₂) ∈ C × C and w _i ∈ T(x _i ), i = 1, 2, where Ξ = {r _n : n ≥ 1}, then for $\hat{x}=P_{F\left(S\right)\cap {\left(GEP\right)}_s\left(\Phi ,\phi \right)}f\left(\hat{x}\right)$, there exists $\hat{w}\in T\left(\hat{x}\right)$ such that $\left(\hat{x},\hat{w}\right)$ is a solution of (GEP) and

$$x_n\to \hat{x},w_n\to \hat{w}and{u}_n\to \hat{x}asn\to \infty .$$

In 2011, Kangtunyakarn [18] proved the following theorem for strict pseudocontractive mapping in Hilbert space by using hybrid method as follows:

Theorem 1.2. Let C be a nonempty closed convex subset of a Hilbert space H. Let F and G be bifunctions from C × C into ℝ satisfying (A₁)-(A₄), respectively. Let A : C → H be a α-inverse strongly monotone mapping and let B : C → H be a β-inverse strongly monotone mapping. Let T : C → C be a κ-strict pseudo-contraction mapping with $F=F\left(T\right)\cap EP\left(F,A\right)\cap EP\left(G,B\right)\ne \varnothing$. Let {x _n } be a sequence generated by x₁ ∈ C = C₁ and

$$\left\{\begin{array}{c}F\left(u_n,u\right)+\left(A{x}_n,u-u_n\right)+\frac{1}{r_n}⟨u-u_n,u_n-x_n⟩\ge 0,\forall u\in C,\hfill \\ G\left(v_n,v\right)+\left(B{x}_n,v-v_n\right)+\frac{1}{s_n}⟨v-v_n,v_n-x_n⟩\ge 0,\forall v\in C,\hfill \\ z_n={\delta }_n{u}_n+\left(1-{\delta }_n\right)v_n\hfill \\ y_n={\alpha }_n{z}_n+\left(1-{\alpha }_n\right)T{z}_n\hfill \\ C_{n+1}=\left\{z\in C_n:∥y_n-z∥\le ∥x_n-z∥\right\},\hfill \\ x_{n+1}=P_{C_{n+1}}{x}_1,\forall n\ge 1,\hfill \end{array}\right.$$

(1.9)

where ${\left\{{\alpha }_n\right\}}_{n=0}^{\infty }$ is sequence in [0,1], r _n ∈ [a, b] ⊂ (0, 2α) and s _n ⊂ [c, d] ⊂ (0, 2β) satisfy the following condition:

$$\begin{array}{c}\left(i\right)\underset{n\to \infty }{\text{lim}}{\delta }_n=\delta \in \left(0,1\right)\\ \left(ii\right)0\le \kappa \le {\alpha }_n<1,\forall n\ge 1\end{array}$$

Then x _n converges strongly to $P_F{x}_1$.

From motivation of (1.7) and (1.9), we define the following algorithm as follows:

Algorithm 1.3. Let T _i , i = 1,2,...,N, be κ _i -pseudo contraction mappings of C into itself and κ = max{κ _i : i = 1,2,..., N} and let S _n be the S-mappings generated by T₁, T₂, ..., T _N and ${\alpha }_1^{\left(n\right)},{\alpha }_2^{\left(n\right)},...,{\alpha }_N^{\left(n\right)}$ where ${\alpha }_j^{\left(n\right)}=\left({\alpha }_1^{n,j},{\alpha }_2^{n,j},{\alpha }_3^{n,j}\right)\in I\times I\times I,I=\left[0,1\right],{\alpha }_1^{n,j}+{\alpha }_2^{n,j}+{\alpha }_3^{n,j}=1$ and $\kappa <a\le {\alpha }_1^{n,j},{\alpha }_3^{n,j}\le b<1$ for all $j=1,2,...,N-1,\kappa <c\le {\alpha }_1^{n,N}\le 1,\kappa \le {\alpha }_3^{n,N}\le d<1,\kappa \le {\alpha }_2^{n,j}\le e<1$ for all j = 1,2,...,N. Let x₁ ∈ C = C₁ and $w_1^1\in T\left(x_1\right),w_1^2\in D\left(x_1\right)$, there exists sequence $\left\{w_n^1\right\},\left\{w_n^2\right\}\in H$ and {x _n }, {u _n }, {v _n } ⊆ C such that

$$\left\{\begin{array}{c}{w}_n^1\in T\left(x_n\right),∥w_n^1-w_{n+1}^1∥\le \left(1+\frac{1}{n}\right)\mathscr{H}\left(T\left(x_n\right),T\left(x_{n+1}\right)\right),\hfill \\ w_n^2\in D\left(x_n\right),∥w_n^2-w_{n+1}^2∥\le \left(1+\frac{1}{n}\right)\mathscr{H}\left(D\left(x_n\right),D\left(x_{n+1}\right)\right),\hfill \\ {\Phi }_1\left(w_n^1,u_n,u\right)+{\phi }_1\left(u\right)-{\phi }_1\left(u_n\right)+\frac{1}{r_n}⟨u_n-x_n,u-u_n⟩\ge 0,\forall u\in C,\hfill \\ {\Phi }_2\left(w_n^2,v_n,v\right)+{\phi }_2\left(v\right)-{\phi }_2\left(v_n\right)+\frac{1}{s_n}⟨v_n-x_n,v-v_n⟩\ge 0,\forall v\in C,\hfill \\ z_n={\delta }_n{P}_C\left(I-\lambda A\right)u_n+\left(1-{\delta }_n\right)P_C\left(I-\eta B\right)v_n,\hfill \\ y_n={\alpha }_n{z}_n+\left(1-{\alpha }_n\right)S_n{z}_n,\hfill \\ C_{n+1}=\left\{z\in C_n:∥y_n-z∥\le ∥x_n-z∥\right\},\hfill \\ x_{n+1}=P_{C_{n+1}}{x}_1,\forall n\ge 1.\hfill \end{array}\right.$$

(1.10)

where D, T : C → CB(H) are $\mathscr{H}$-Lipschitz continuous with constant μ₁, μ₂, respectively, Φ₁, Φ₂ : H × C × C → ℝ are equilibrium-like functions satisfying (H 1)-(H 3), A : C → H is a α-inverse strongly monotone mapping and B : C → H is a β-inverse strongly monotone mapping.

In this article, we prove under some control conditions on {δ _n }, {α _n }, {s _n }, and {r _n } that the sequence {x _n } generated by (1.7) converges strongly to $P_F{x}_1$ where $F={\cap }_{i=1}^{N}F\left(T_i\right)\cap {\left(GEP\right)}_s\left({\Phi }_1,{\phi }_1\right)\cap {\left(GEP\right)}_s\left({\Phi }_2,{\phi }_2\right)\cap F\left(G_1\right)\cap F\left(G_2\right)$, G₁, G₂ : C → C are defined by G₁(x) = P _C (x - λAx), G₂(x) = P _C (x - ηBx), ∀x ∈ C and $P_F{x}_1$ is solution of the following system of variational inequality:

$$\left\{\begin{array}{c}⟨A{x}^{*},x-x^{*}⟩\ge 0,\hfill \\ ⟨B{x}^{*},x-x^{*}⟩\ge 0.\hfill \end{array}\right.$$

2 Preliminaries

In this section, we need the following lemmas and definition to prove our main result.

Let C be a nonempty closed convex subset of H. Then for any x ∈ H, there exists a unique nearest point in C, denoted by P _C x, such that

$$∥x-P_{C}x∥\le ∥x-y∥,\text{for}\text{all}y\in C.$$

The following lemma is a property of P _C .

Lemma 2.1. (See [19].) Given x ∈ H and y ∈ C. Then P _C x = y if and only if there holds the inequality

$$⟨x-y,y-z⟩\ge 0\forall z\in C.$$

Lemma 2.2. (See [20] ) Let C be a closed convex subset of a strictly convex Banach space E. Let {T _n : n ∈ ℕ} be a sequence of nonexpansive mappings on C. Suppose ${\cap }_{n=1}^{\infty }F\left(T_n\right)$ is nonempty. Let {λ _n } be a sequence of positive numbers with ${\sum }_{n=1}^{\infty }{\lambda }_n=1$. Then a mapping S on C defined by

$$S\left(x\right)={\sum }_{n=1}^{\infty }{\lambda }_n{T}_{n}x$$

for x ∈ C is well defined, nonexpansive and $F\left(S\right)={\cap }_{n=1}^{\infty }F\left(T_n\right)$ hold.

The following lemma is well known.

Lemma 2.3. Let H be Hilbert space, C be a nonempty closed convex subset of H. Let T : C → C be a κ-strictly pseudo-contractive, then the fixed point set F(T) of T is closed and convex so that the projection P_F(T)is well defined.

In 2009, Kangtunyakarn and Suantai [21] introduced the S-mapping generated by a finite family of κ-strictly pseudo contractive mappings and real numbers as follows:

Definition 2.1. Let C be a nonempty convex subset of real Hilbert space. Let ${\left\{T_i\right\}}_{i=1}^N$ be a finite family of κ _i -strict pseudo-contractions of C into itself. For each j = 1,2,..., N, let ${\alpha }_j=\left(a_1^j,{\alpha }_2^j,{\alpha }_3^j\right)\in I\times I\times I$ where I ∈ [0,1] and ${\alpha }_1^j+{\alpha }_2^j+{\alpha }_3^j=1$. We define the mapping S : C → C as follows:

$$\begin{array}{ll}\hfill U_0& =I\\ \hfill U_1& ={\alpha }_1^1{T}_1{U}_0+{\alpha }_2^1{U}_0+{\alpha }_3^{1}I\\ \hfill U_2& ={\alpha }_1^2{T}_2{U}_1+{\alpha }_2^2{U}_1+{\alpha }_3^{2}I\\ \hfill U_3& ={\alpha }_1^3{T}_3{U}_2+{\alpha }_2^3{U}_2+{\alpha }_3^{3}I\\ \cdot \\ \cdot \\ \cdot \\ \hfill U_{N-1}& ={\alpha }_1^{N-1}{T}_{N-1}{U}_{N-2}+{\alpha }_2^{N-1}{U}_{N-2}+{\alpha }_3^{N-1}I\\ \hfill S& =U_N={\alpha }_1^N{T}_N{U}_{N-1}+{\alpha }_2^N{U}_{N-1}+{\alpha }_3^{N}I.\end{array}$$

(2.1)

This mapping is called S-mapping generated by T₁, ..., T _N and α₁, α₂, ..., α _N .

Lemma 2.4. (See [21] ) Let C be a nonempty closed convex subset of real Hilbert space. Let ${\left\{T_i\right\}}_{i=1}^N$ be a finite family of κ-strict pseudo contraction mapping of C into C with ${\cap }_{i=1}^{N}F\left(T_i\right)\ne \mathrm{0̸}$ and κ = max{κ _i : i = 1, 2,..., N} and let ${\alpha }_j=\left(a_1^j,{\alpha }_2^j,{\alpha }_3^j\right)\in I\times I\times I$, j = 1,2,3,...,N, where $I=\left[0,1\right],{\alpha }_1^j+{\alpha }_2^j+{\alpha }_3^j=1,{\alpha }_1^j,{\alpha }_3^j\in \left(\kappa ,1\right)$ for all j = 1,2,...,N - 1 and ${\alpha }_1^N\in \left(\kappa ,1\right],{\alpha }_3^N\in \left[\kappa ,1\right){\alpha }_2^j\in \left[\kappa ,1\right)$ for all j = 1,2,..., N. Let S be the mapping generated by T₁,....,T _N and α₁, α₂,...,α _N . Then $F\left(S\right)={\cap }_{i=1}^{N}F\left(T_i\right)$ and S is a nonexpansive mapping.

Lemma 2.5. (See [22] ) Let C be a nonempty closed convex subset of a real Hilbert space H and S : C → C be a self-mapping of C. If S is a κ-strict pseudo-contraction mapping, then S satisfies the Lipschitz condition

$$∥Sx-Sy∥\le \frac{1+\kappa }{1-\kappa }∥x-y∥,\forall x,y\in C.$$

We prove the following lemma by using the concept of the S-mapping as follows:

Lemma 2.6. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T _i , i = 1,2,...,N be κ _i strictly pseudo-contraction mappings of C into itself and κ = max{κ _i : i = 1,2,...,N} and let ${\alpha }_j^{\left(n\right)}=\left({\alpha }_1^{n,j},{\alpha }_2^{n,j},{\alpha }_3^{n,j}\right),{\alpha }_j=\left({\alpha }_1^j,{\alpha }_2^j,{\alpha }_3^j\right)\in I\times I\times I$, where $I=\left[0,1\right],{\alpha }_1^{n,j}+{\alpha }_2^{n,j}+{\alpha }_3^{n,j}=1$ and ${\alpha }_1^j+{\alpha }_2^j+{\alpha }_3^j=1$ such that ${\alpha }_i^{n,j}\to {\alpha }_i^j\in \left[0,1\right]$ as n → ∞ for i = 1, 3 and j = 1,2,3,..., N. For every n ∈ ℕ, let S and S _n be the S-mapping generated by T₁, T₂,..., T _N and α₁, α₂,...,α _N and T₁, T₂,..., T _N and ${\alpha }_1^{\left(n\right)},{\alpha }_2^{\left(n\right)},\dots ,{\alpha }_N^{\left(n\right)}$, respectively. Then lim_n→∞∥S _n x _n - Sx _n ∥ = 0 for every bounded sequence {x _n } in C.

Proof. Let {x _n } be bounded sequence in C, U _k and U_n,kbe generated by T₁,T₂,...,T _N and α₁,α₂,...,α _N and T₁,T₂,...,T _N and ${\alpha }_1^{\left(n\right)},{\alpha }_2^{\left(n\right)},\dots ,{\alpha }_N^{\left(n\right)}$, respectively. For each n ∈ ℕ, we have

$$\begin{array}{ll}\hfill ∥U_{n,1}{x}_n-U_1{x}_n∥& =∥{\alpha }_1^{n,1}{T}_1{x}_n+\left(1-{\alpha }_1^{n,1}\right)x_n-{\alpha }_1^1{T}_1{x}_n-\left(1-{\alpha }_1^1\right)x_n∥\\ =∥{\alpha }_1^{n,1}{T}_1{x}_n-{\alpha }_1^{n,1}{x}_n-{\alpha }_1^1{T}_1{x}_n+{\alpha }_1^1{x}_n∥\\ =∥\left({\alpha }_1^{n,1}-{\alpha }_1^1\right)T_1{x}_n-\left({\alpha }_1^{n,1}-{\alpha }_1^1\right)x_n∥\\ =\left|a_1^{n,1}-{\alpha }_1^1\right|∥T_1{x}_n-x_n∥\end{array}$$

(2.2)

and for k ∈ {2, 3,..., N}, by using Lemma 2.5, we obtain

$$\begin{array}{ll}\hfill ∥U_{n,k}{x}_n-U_k{x}_n∥& =∥{\alpha }_1^{n,k}{T}_k{U}_{n,k-1}{x}_n+{\alpha }_2^{n,k}{U}_{n,k-1}{x}_n+{\alpha }_3^{n,k}{x}_n\\ -{\alpha }_1^k{T}_k{U}_{k-1}{x}_n-{\alpha }_2^k{U}_{k-1}{x}_n-{\alpha }_3^k{x}_n∥\\ =∥{\alpha }_1^{n,k}{T}_k{U}_{n,k-1}{x}_n+{\alpha }_3^{n,k}{x}_n-{\alpha }_1^k{T}_k{U}_{k-1}{x}_n-{\alpha }_3^k{x}_n\\ +{\alpha }_2^{n,k}{U}_{n,k-1}{x}_n-{\alpha }_2^k{U}_{k-1}{x}_n∥\\ =∥{\alpha }_1^{n,k}{T}_k{U}_{n,k-1}{x}_n-{\alpha }_1^{n,k}{T}_k{U}_{k-1}{x}_n+{\alpha }_1^{n,k}{T}_k{U}_{k-1}{x}_n\\ -{\alpha }_1^k{T}_k{U}_{k-1}{x}_n+\left({\alpha }_3^{n,k}-{\alpha }_3^k\right)x_n+{\alpha }_2^{n,k}{U}_{n,k-1}{x}_n-{\alpha }_2^k{U}_{k-1}{x}_n∥\\ =∥{\alpha }_1^{n,k}\left(T_k{U}_{n,k-1}{x}_n-T_k{U}_{k-1}{x}_n\right)+\left({\alpha }_1^{n,k}-{\alpha }_1^k\right)T_k{U}_{k-1}{x}_n\\ +\left({\alpha }_3^{n,k}-{\alpha }_3^k\right)x_n+{\alpha }_2^{n,k}{U}_{n,k-1}{x}_n-{\alpha }_2^k{U}_{k-1}{x}_n∥\\ =∥{\alpha }_1^{n,k}\left(T_k{U}_{n,k-1}{x}_n-T_k{U}_{k-1}{x}_n\right)+\left({\alpha }_1^{n,k}-{\alpha }_1^k\right)T_k{U}_{k-1}{x}_n\\ +\left({\alpha }_3^{n,k}-{\alpha }_3^k\right)x_n+{\alpha }_2^{n,k}{U}_{n,k-1}{x}_n-{\alpha }_2^{n,k}{U}_{k-1}{x}_n\\ +{\alpha }_2^{n,k}{U}_{k-1}{x}_n-{\alpha }_2^k{U}_{k-1}{x}_n∥\\ =∥{\alpha }_1^{n,k}\left(T_k{U}_{n,k-1}{x}_n-T_k{U}_{k-1}{x}_n\right)+\left({\alpha }_1^{n,k}-{\alpha }_1^k\right)T_k{U}_{k-1}{x}_n\\ +\left({\alpha }_3^{n,k}-{\alpha }_3^k\right)x_n+{\alpha }_2^{n,k}\left(U_{n,k-1}{x}_n-U_{k-1}{x}_n\right)\\ +\left({\alpha }_2^{n,k}-{\alpha }_2^k\right)U_{k-1}{x}_n∥\\ \le {\alpha }_1^{n,k}∥T_k{U}_{n,k-1}{x}_n-T_k{U}_{k-1}{x}_n∥+\left|{\alpha }_1^{n,k}-{\alpha }_1^k\right|∥T_k{U}_{k-1}{x}_n∥\\ +\left|{\alpha }_3^{n,k}-{\alpha }_3^k\right|∥x_n∥+{\alpha }_2^{n,k}∥U_{n,k-1}{x}_n-U_{k-1}{x}_n∥\\ +\left|{\alpha }_2^{n,k}-{\alpha }_2^k\right|∥U_{k-1}{x}_n∥\\ ={\alpha }_1^{n,k}∥T_k{U}_{n,k-1}{x}_n-T_k{U}_{k-1}{x}_n∥+\left|{\alpha }_1^{n,k}-{\alpha }_1^k\right|∥T_k{U}_{k-1}{x}_n∥\\ +{\alpha }_2^{n,k}∥U_{n,k-1}{x}_n-U_{k-1}{x}_n∥+\left|1-{\alpha }_1^{n,k}-{\alpha }_3^{n,k}-1\right.\\ \left.+{\alpha }_1^k+{\alpha }_3^k\right|∥U_{k-1}{x}_n∥+\left|{\alpha }_3^{n,k}-{\alpha }_3^k\right|∥x_n∥\\ \le {\alpha }_1^{n,k}\frac{1+\kappa }{1-\kappa }∥U_{n,k-1}{x}_n-U_{k-1}{x}_n∥+\left|{\alpha }_1^{n,k}\right.\\ \left.-{\alpha }_1^k\right|∥T_k{U}_{k-1}{x}_n∥+{\alpha }_2^{n,k}∥U_{n,k-1}{x}_n-U_{k-1}{x}_n∥+\left(\left|{\alpha }_1^k-{\alpha }_1^{n,k}\right|\right.\\ \left.+\left|{\alpha }_3^{n,k}-{\alpha }_3^k\right|\right)∥U_{k-1}{x}_n∥+\left|{\alpha }_3^{n,k}-{\alpha }_3^k\right|∥x_n∥\\ \le \frac{1+\kappa }{1-\kappa }∥U_{n,k-1}{x}_n-U_{k-1}{x}_n∥+\left|{\alpha }_1^{n,k}-{\alpha }_1^k\right|∥T_k{U}_{k-1}{x}_n∥\\ +\frac{1-\kappa }{1-\kappa }∥U_{n,k-1}{x}_n-U_{k-1}{x}_n∥+\left(\left|{\alpha }_1^k-{\alpha }_1^{n,k}\right|\right.\\ \left.+\left|{\alpha }_3^{n,k}-{\alpha }_3^k\right|\right)∥U_{k-1}{x}_n∥+\left|{\alpha }_3^{n,k}-{\alpha }_3^k\right|∥x_n∥\\ \le \frac{2}{1-\kappa }∥U_{n,k-1}{x}_n-U_{k-1}{x}_n∥+\left|{\alpha }_1^{n,k}-{\alpha }_1^k\right|\left(∥T_k{U}_{k-1}{x}_n∥+∥U_{k-1}{x}_n∥\right)\\ +\left|{\alpha }_3^{n,k}-{\alpha }_3^k\right|\left(∥U_{k-1}{x}_n∥+∥x_n∥\right).\end{array}$$

(2.3)

By (2.2) and (2.3), we have

$$\begin{array}{ll}\hfill ∥S_n{x}_n-S{x}_n∥& =∥U_{n,N}{x}_n-U_N{x}_n∥\\ \le \frac{2}{1-\kappa }∥U_{n,N-1}{x}_n-U_{N-1}{x}_n∥+\left|{\alpha }_1^{n,N}-{\alpha }_1^N\right|\left(∥T_N{U}_{N-1}{x}_n∥\right.\\ \left.+∥U_{N-1}{x}_n∥\right)+\left|{\alpha }_3^{n,N}-{\alpha }_3^N\right|\left(∥U_{N-1}{x}_n∥+∥x_n∥\right)\\ \le \frac{2}{1-\kappa }\left(\frac{2}{1-\kappa }∥U_{n,N-2}{x}_n-U_{N-2}{x}_n∥+\left|{\alpha }_1^{n,N-1}\right.\right.\\ \left.-{\alpha }_1^{N-1}\right|\left(∥T_{N-1}{U}_{N-2}{x}_n∥\right.\\ \left.\left.+∥U_{N-2}{x}_n∥\right)+\left|{\alpha }_3^{n,N-1}-{\alpha }_3^{N-1}\right|\left(∥U_{N-2}{x}_n∥+∥x_n∥\right)\right)\\ +\left|{\alpha }_1^{n,N}-{\alpha }_1^N\right|\left(∥T_N{U}_{N-1}{x}_n∥+∥U_{N-1}{x}_n∥\right)\\ +\left|{\alpha }_3^{n,N}-{\alpha }_3^N\right|\left(∥U_{N-1}{x}_n∥+∥x_n∥\right)\\ ={\left(\frac{2}{1-\kappa }\right)}^2∥U_{n,N-2}{x}_n-U_{N-2}{x}_n∥+\sum _{j=N-1}^N{\left(\frac{2}{1-\kappa }\right)}^{N-j}\left|{\alpha }_1^{n,j}\right.\\ \left.-{\alpha }_1^j\right|\left(∥T_j{U}_{j-1}{x}_n∥+∥U_{j-1}{x}_n∥\right)\\ +\sum _{j=N-1}^N{\left(\frac{2}{1-\kappa }\right)}^{N-j}\left|{\alpha }_3^{n,j}-{\alpha }_3^j\right|\left(∥U_{j-1}{x}_n∥+∥x_n∥\right)\\ \cdot \\ \cdot \\ \cdot \\ \cdot \\ \le {\left(\frac{2}{1-\kappa }\right)}^{N-1}∥U_{n,1}{x}_n-U_1{x}_n∥+\sum _{j=2}^N{\left(\frac{2}{1-\kappa }\right)}^{N-j}\left|{\alpha }_1^{n,j}\right.\\ \left.-{\alpha }_1^j\right|\left(∥T_j{U}_{j-1}{x}_n∥+∥U_{j-1}{x}_n∥\right)+\sum _{j=2}^N{\left(\frac{2}{1-\kappa }\right)}^{N-j}\left|{\alpha }_3^{n,j}-{\alpha }_3^j\right|\left(∥U_{j-1}{x}_n∥+∥x_n∥\right)\\ ={\left(\frac{2}{1-\kappa }\right)}^{N-1}\left|{\alpha }_1^{n,1}-{\alpha }_1^1\right|∥T_1{x}_n-x_n∥+\sum _{j=2}^N{\left(\frac{2}{1-\kappa }\right)}^{N-j}\left|{\alpha }_1^{n,j}\right.\\ \left.-{\alpha }_1^j\right|\left(∥T_j{U}_{j-1}{x}_n∥+∥U_{j-1}{x}_n∥\right)\\ +\sum _{j=2}^N{\left(\frac{2}{1-\kappa }\right)}^{N-j}\left|{\alpha }_3^{n,j}-{\alpha }_3^j\right|\left(∥U_{j-1}{x}_n∥+∥x_n∥\right).\end{array}$$

This together with the assumption ${\alpha }_i^{n,j}\to {\alpha }_i^j$ as n → ∞ (i = 1, 3, j = 1,2,..., N), we can conclude that

$$\underset{n\to \infty }{\text{lim}}∥S_n{x}_n-S{x}_n∥=0.$$

Lemma 2.7. (See [23]) Let E be a uniformly convex Banach space, C be a nonempty closed convex subset of E and S : C → C be a nonexpansive mapping. Then I - S is demi-closed at zero.

Lemma 2.8. (See [24]) Let C be a closed convex subset of H. Let {x _n } be a sequence in H and u ∈ H. Let q = P _C u, if {x _n } is such the ω(x _n ) ⊂ C and satisfy the condition

$$∥x_n-u∥\le ∥u-q∥,\forall n\in \mathbb{N}.$$

Then x _n → q, as n → ∞.

Definition 2.2. A multivalued map T : C → CB(H) is say to be $\mathscr{H}$-Lipschitz continuous if there exists a constant μ > 0 such that

$$\mathscr{H}\left(T\left(u\right)-T\left(v\right)\right)\le \mu ∥u-v∥,\forall u,v\in C,$$

where $\mathscr{H}\left(.,.\right)$ is the Hausdorff metric on CB(H).

Lemma 2.9. (Nadler's theorem, see [25]) Let (X, ∥ ⋅ ∥) be a normed vector space and $\mathscr{H}\left(.,.\right)$ is the Hausdorff metric on CB(H). If U, V ∈ CB(X), then for any given ϵ > 0 and u ∈ U, there exists v ∈ V such that

$$∥u-v∥\le \left(1+\epsilon \right)\mathscr{H}\left(U,V\right).$$

Let C be a nonempty closed convex subset of a real Hilbert space H. Let φ: C → H be a real-valued function, T : C → CB(H) be a multivalued map and Φ : H × C × C → ℝ be an equilibrium-like function.

To solve the GEP, let us assume that the equilibrium-like function Φ : H × C × C → ℝ satisfies the following conditions with respect to the multivalued map T: C → CB(H).

(H 1) For each fixed v ∈ C, (ω, u) ↦ Φ(ω, u, v) is an upper semicontinuous function from H × C to ℝ, that is, for (ω, u) ∈ H × C, whenever ω _n → ω and u _n → u as n → ∞,

$$\underset{n\to \infty }{\text{lim sup}}\Phi \left({\omega }_n,u_n,v\right)\le \Phi \left(\omega ,u,v\right);$$

(H 2) For each fixed (w, v) ∈ H × C, u ↦ Φ(w, u, v) is a concave function;

(H 3) For each fixed (w, u) ∈ H × C, v ↦ Φ(w, u, v) is a convex function.

Theorem 2.10. (See [17]) Let C be a nonempty, bounded, closed, and convex subset of a real Hilbert space H, and let φ : C → ℝ be a lower semicontinuous and convex functional. Let T : C → CB(H) be $\mathscr{H}$-Lipschitz continuous with constant μ, and Φ : H × C × C → ℝ be an equilibrium-like function satisfying (H 1)-(H 3). Let r > 0 be a constant. For each x ∈ C, take w _x ∈ T(x) arbitrarily and define a mapping T _r : C → C as follows:

$$T_r\left(x\right)=\left\{u\in C:\Phi \left(w_x,u,v\right)+\phi \left(v\right)-\phi \left(u\right)+\frac{1}{r}⟨u-x,v-u⟩\ge 0,\forall v\in C\right\}.$$

Then, there hold the following:

(a) T _r is single-valued;

(b) T _r is firmly nonexpansive (that is, for any u, v ∈ C, ∥T _r u - T _r v∥² ≤ 〈T _r u-T _r v, u-v〉) if

$$\Phi \left(w_1,T_r\left(x_1\right),T_r\left(x_2\right)\right)+\Phi \left(w_2,T_r\left(x_2\right),T_r\left(x_1\right)\right)\le 0,$$

for all (x₁, x₂) ∈ C × C and all w _i ∈ T(x_i), i = 1,2;

(c) F(T _r ) = (GEP) _s (Φ, φ)

(d) (GEP) _s (Φ, φ) is closed and convex.

Lemma 2.11. (See [26]) Let C be a nonempty closed convex subset of a Hilbert space H and let G : C → C be defined by

$$G\left(x\right)=P_C\left(x-\lambda Ax\right),\forall x\in C,$$

with ∀λ > 0. Then x* ∈ VI (C, A) if and only if x* ∈ F(G).

3 Main results

In this section, we prove a strong convergence theorem of the sequence {x _n } generated by (1.10) to $P_F{x}_1$.

Theorem 3.1. Let C be a nonempty bounded, closed, and convex subset of Hilbert space H and let φ₁, φ₂ : be a lower semicontinuous and convex function. Let D, T : C → CB(H) be $\mathscr{H}$-Lipschitz continuous with constant μ₁, μ₂, respectively, Φ₁,Φ₂: H × C × C→ ℝ be equilibrium-like functions satisfying (H 1) - (H3). Let A: C → H be a α-inverse strongly monotone mapping and B : C → H be a β-inverse strongly monotone mapping, let T _i , i = 1,2,...,N, be κ _i -pseudo contraction mappings of C into itself and κ = max{κ _i :i = 1,2,..., N} with $F={\cap }_{i=1}^{N}F\left(T_i\right)\cap {\left(GEP\right)}_s\left({\Phi }_1,{\phi }_1\right)\cap {\left(GEP\right)}_s\left({\Phi }_2,{\phi }_2\right)\cap F\left(G_1\right)\cap F\left(G_2\right)$, where G₁, G₂ : C → C are defined by G₁(x) P _C (x-λAx), G₂(x) = P _C (x-ηBx), ∀x ∈ C. Let S _n be the S-mappings generated by T₁,T₂,...,T _N and ${\alpha }_1^{\left(n\right)},{\alpha }_2^{\left(n\right)},...,{\alpha }_N^{\left(n\right)}$ where ${\alpha }_j^{\left(n\right)}=\left({\alpha }_1^{n,j},{\alpha }_2^{n,j},{\alpha }_3^{n,j}\right)\in I\times I\times I,I=\left[0,1\right],{\alpha }_1^{n,j}+{\alpha }_2^{n,j}+{\alpha }_3^{n,j}=1$ and $\kappa <a\le {\alpha }_1^{n,j},{\alpha }_3^{n,j}\le b<1$ for all $j=1,2,...,N-1,\kappa <c\le {\alpha }_1^{n,N}\le 1,\kappa \le {\alpha }_3^{n,N}\le d<1,\kappa \le {\alpha }_2^{n,j}\le e<1$ for all j = 1,2,...,N and let {x _n }, {u _n }, {v _n }, $\left\{w_n^1\right\}$, and $\left\{w_n^2\right\}$ be sequences generated by (1.10), where {α _n } is a sequence in [0,1], r _n , λ ∈ [a, b] ⊂ (0, 2α) and s _n , η ∈ [c, d] ⊂ (0, 2β), for every n ∈ ℕ and suppose the following conditions hold:

(i) $\underset{n\to \infty }{\text{lim}}{\delta }_n=\delta \in \left(0,1\right)$,

(ii) 0 ≤ κ ≤ α _n < 1, ∀n ≥ 1,

(iii) $\sum _{n=1}^{\infty }\left|{\alpha }_1^{n+1,j}-{\alpha }_1^{n,j}\right|<\infty ,\sum _{n=1}^{\infty }\left|{\alpha }_3^{n+1,j}-{\alpha }_3^{n,j}\right|<\infty$, for all j ∈ {1,2,3,...,N}.

(iv) There exists λ₁, λ₂ such that

$$\left\{\begin{array}{c}{\Phi }_1\left(w_1^1,T_{r_1}\left(x_1\right),T_{r_2}\left(x_2\right)\right)+{\Phi }_1\left(w_2^1,T_{r_2}\left(x_2\right),T_{r_1}\left(x_1\right)\right)\le -{\lambda }_1{∥T_{r_1}\left(x_1\right)-T_{r_2}\left(x_2\right)∥}^{2}and\hfill \\ {\Phi }_2\left(w_1^2,T_{s_1}\left(x_1\right),T_{s_2}\left(x_2\right)\right)+{\Phi }_2\left(w_2^2,T_{s_2}\left(x_2\right),T_{s_1}\left(x_1\right)\right)\le -{\lambda }_2{∥T_{s_1}\left(x_1\right)-T_{s_2}\left(x_2\right)∥}^2\hfill \end{array}\right.$$

(3.1)

for all $\left(r_1,r_2\right)\in \Theta \times \Theta ,\left(s_1,s_2\right)\in \Xi \times \Xi ,w_i^1\in T\left(x_i\right)$ and $w_i^2\in D\left(x_i\right)$, for i = 1,2 where Θ = {r _n : n ≥ 1} and Ξ = {s _n : n ≥ 1}. Then {x _n } converges strongly to $P_F{x}_1$ which is a solution of (3.2):

$$\left\{\begin{array}{c}⟨A{x}^{*},x-x^{*}⟩\ge 0,\hfill \\ ⟨B{x}^{*},x-x^{*}⟩\ge 0.\hfill \end{array}\right.$$

(3.2)

Proof. From (3.1) for every r ∈ Θ, we have

$${\Phi }_1\left(w_1^1,T_r\left(x_1\right),T_r\left(x_2\right)\right)+{\Phi }_1\left(w_2^1,T_r\left(x_2\right),T_r\left(x_1\right)\right)\le -{\lambda }_1{∥T_r\left(x_1\right)-T_r\left(x_2\right)∥}^2\le 0,$$

(3.3)

for all (x₁, x₂) ∈ C × C and $w_i^1\in T\left(x_i\right),i=1,2$.

Similarly, for every s ∈ Ξ, we have

$${\Phi }_2\left(w_1^2,T_s\left(x_1\right),T_s\left(x_2\right)\right)+{\Phi }_2\left(w_2^2,T_s\left(x_2\right),T_s\left(x_1\right)\right)\le -{\lambda }_2{∥T_s\left(x_1\right)-T_s\left(x_2\right)∥}^2\le 0.$$

(3.4)

for all (x₁, x₂) ∈ C × C and $w_i^2\in D\left(x_i\right),i=1,2$. From (3.3) and (3.4), we have Theorem 2.10 hold.

It is easy to see that I - λ A and I - η B are nonexpansive mapping. Indeed, since A is a α-inverse strongly monotone mapping with λ ∈ (0, 2α), we have

$$\begin{array}{ll}\hfill {∥\left(I-\lambda A\right)x-\left(I-\lambda A\right)y∥}^2& ={∥x-y-\lambda \left(Ax-Ay\right)∥}^2\\ ={∥x-y∥}^2-2\lambda ⟨x-y,Ax-Ay⟩+{\lambda }^2{∥Ax-Ay∥}^2\\ \le {∥x-y∥}^2-2\alpha \lambda {∥Ax-Ay∥}^2+{\lambda }^2{∥Ax-Ay∥}^2\\ ={∥x-y∥}^2+\lambda \left(\lambda -2\alpha \right){∥Ax-Ay∥}^2\\ \le {∥x-y∥}^2.\end{array}$$

Thus (I - λA) is nonexpansive, so is I - ηB. Since

$${\Phi }_1\left(w_n^1,u_n,u\right)+{\phi }_1\left(u\right)-{\phi }_1\left(u_n\right)+\frac{1}{r_n}⟨u_n-x_n,u-u_n⟩\ge 0,\forall u\in C,$$

and Theorem 2.10, we have $u_n=T_{r_n}{x}_n$. Since

$${\Phi }_2\left(w_n^2,v_n,v\right)+{\phi }_2\left(v\right)-{\phi }_2\left(v_n\right)+\frac{1}{s_n}⟨v_n-x_n,v-v_n⟩\ge 0,\forall v\in C,$$

and Theorem 2.10, we have $v_n=T_{s_n}{x}_n$. Let $z\in F$, again by Theorem 2.10, we have $z=T_{r_n}z=T_{s_n}z=P_C\left(I-\lambda A\right)z=P_C\left(I-\eta B\right)z$. From nonexpansiveness of $\left\{T_{r_n}\right\},\left\{T_{s_n}\right\},\left\{I-\lambda A\right\}$, and {I - ηB}, we have

$$\begin{array}{ll}\hfill ∥z_n-z∥& =∥{\delta }_n\left(P_C\left(I-\lambda A\right)u_n-z\right)+\left(1-{\delta }_n\right)\left(P_C\left(I-\eta B\right)v_n-z\right)∥\\ \le {\delta }_n∥P_C\left(I-\lambda A\right)u_n-z∥+\left(1-{\delta }_n\right)∥P_C\left(I-\eta B\right)v_n-z∥\\ \le {\delta }_n∥T_{r_n}{x}_n-z∥+\left(1-{\delta }_n\right)∥T_{s_n}{x}_n-z∥\\ \le ∥x_n-z∥.\end{array}$$

(3.5)

By (3.5), we have

$$\begin{array}{ll}\hfill ∥y_n-z∥& =∥{\alpha }_n\left(z_n-z\right)+\left(1-{\alpha }_n\right)\left(S_n{z}_n-z\right)∥\\ \le {\alpha }_n∥z_n-z∥+\left(1-{\alpha }_n\right)∥S_n{z}_n-z∥\\ \le ∥z_n-z∥\le ∥x_n-z∥.\end{array}$$

(3.6)

Next, we show that C _n is closed and convex for every n ∈ ℕ. It is obvious that C _n is closed. In fact, we know that, for z ∈ C _n ,

$$∥y_n-z∥\le ∥x_n-z∥\text{is}\text{equivalent}\text{to}{∥y_n-x_n∥}^2+2⟨y_n-x_n,x_n-z⟩\le 0.$$

So, we have that ∀z₁, z₂ ∈ C _n and t ∈ (0,1), it follows that

$$\begin{array}{ll}\hfill {∥y_n-x_n∥}^2& +2⟨y_n-x_n,x_n-\left(t{z}_1+\left(1-t\right)z_2\right)⟩\\ =t\left(2⟨y_n-x_n,x_n-z_1⟩+{∥y_n-x_n∥}^2\right)\\ +\left(1-t\right)\left(2⟨y_n-x_n,x_n-z_2⟩+{∥y_n-x_n∥}^2\right)\\ \le 0,\end{array}$$

then, we have C _n is convex. By Theorem 2.10 and Lemma 2.3, we conclude that $F$ is closed and convex. This implies that $P_F$ is well defined. Next, we show that $F\subset C_n$ for every n ∈ ℕ. Putting $q\in F$, by (3.6), it is easy to see that q ∈ C _n , then we have $F\subset C_n$ for all n ∈ ℕ. Since $x_n=P_{C_n}{x}_1$, for every w ∈ C _n , we have

$$∥x_n-x_1∥\le ∥w-x_1∥,\forall n\in \mathbb{N}.$$

(3.7)

In particular, we have

$$∥x_n-x_1∥\le ∥P_F{x}_1-x_1∥.$$

(3.8)

Since C is bounded, we have {x _n } is bounded, so are {u _n }, {v _n }, {z _n }, and {y _n }. Since $x_{n+1}=P_{C_{n+1}}{x}_1\in C_{n+1}\subset C_n$ and $x_n=P_{C_n}{x}_1$, we have

$$\begin{array}{ll}\hfill 0& \le ⟨x_1-x_n,x_n-x_{n+1}⟩\\ =⟨x_1-x_n,x_n-x_1+x_1-x_{n+1}⟩\\ \le -{∥x_n-x_1∥}^2+∥x_n-x_1∥∥x_1-x_{n+1}∥,\end{array}$$

it implies that

$$∥x_n-x_1∥\le ∥x_{n+1}-x_1∥.$$

Hence, we have lim_n→∞∥x _n - x₁∥ exists. Since

$$\begin{array}{ll}\hfill {∥x_n-x_{n+1}∥}^2& ={∥x_n-x_1+x_1-x_{n+1}∥}^2\\ ={∥x_n-x_1∥}^2+2⟨x_n-x_1,x_1-x_{n+1}⟩+{∥x_1-x_{n+1}∥}^2\\ ={∥x_n-x_1∥}^2+2⟨x_n-x_1,x_1-x_n+x_n-x_{n+1}⟩+{∥x_1-x_{n+1}∥}^2\\ ={∥x_n-x_1∥}^2-2{∥x_n-x_1∥}^2+2⟨x_n-x_1,x_n-x_{n+1}⟩+{∥x_1-x_{n+1}∥}^2\\ \le {∥x_1-x_{n+1}∥}^2-{∥x_n-x_1∥}^2,\end{array}$$

(3.9)

it implies that

$$\underset{n\to \infty }{\text{lim}}∥x_n-x_{n+1}∥=0.$$

(3.10)

Since $x_{n+1}=P_{C_{n+1}}{x}_1\in C_{n+1}$, we have

$$∥y_n-x_{n+1}∥\le ∥x_n-x_{n+1}∥,$$

by (3.10), we have

$$\underset{n\to \infty }{\text{lim}}∥y_n-x_{n+1}∥=0.$$

(3.11)

Since

$$∥y_n-x_n∥\le ∥y_n-x_{n+1}∥+∥x_{n+1}-x_n∥,$$

by (3.10) and (3.11), we have

$$\underset{n\to \infty }{\text{lim}}∥y_n-x_n∥=0.$$

(3.12)

Next, we show that

$$\underset{n\to \infty }{\text{lim}}∥z_n-S_n{z}_n∥=0.$$

(3.13)

By definition of y _n , we have

$$y_n-z_n=\left(1-{\alpha }_n\right)\left(S_n{z}_n-z_n\right).$$

(3.14)

Claim that

$$\underset{n\to \infty }{\text{lim}}∥z_n-x_n∥=0.$$

(3.15)

Putting M _n = P _C (I - λA)u _n and N _n = P _C (I - ηB)v _n , we have

$$∥z_n-x_n∥\le {\delta }_n∥M_n-x_n∥+\left(1-{\delta }_n\right)∥N_n-x_n∥.$$

(3.16)

Let $z\in F$. Since $T_{r_n}$ is firmly nonexpansive mapping and $T_{r_n}{x}_n=u_n$, we have

$$\begin{array}{ll}\hfill {∥z-u_n∥}^2& ={∥T_{r_n}z-T_{r_n}{x}_n∥}^2\\ \le ⟨T_{r_n}z-T_{r_n}{x}_n,z-x_n⟩\\ =\frac{1}{2}\left({∥u_n-z∥}^2+{∥x_n-z∥}^2-{∥u_n-x_n∥}^2\right).\end{array}$$

Hence

$${∥u_n-z∥}^2\le {∥x_n-z∥}^2-{∥u_n-x_n∥}^2.$$

(3.17)

Since $T_{r_n}$ is firmly nonexpansive mapping and $T_{s_n}{x}_n=v_n$, by using the same method as (3.17), we have

$${∥v_n-z∥}^2\le {∥x_n-z∥}^2-{∥v_n-x_n∥}^2.$$

(3.18)

By nonexpansiveness of S _n and (3.17), (3.18), we have

$$\begin{array}{ll}\hfill {∥y_n-z∥}^2& \le {∥z_n-z∥}^2\\ \le {\delta }_n{∥u_n-z∥}^2+\left(1-{\delta }_n\right){∥v_n-z∥}^2\\ \le {\delta }_n\left({∥x_n-z∥}^2-{∥u_n-x_n∥}^2\right)+\left(1-{\delta }_n\right)\left({∥x_n-z∥}^2-{∥v_n-x_n∥}^2\right)\\ ={∥x_n-z∥}^2-{\delta }_n{∥u_n-x_n∥}^2-\left(1-{\delta }_n\right){∥v_n-x_n∥}^2,\end{array}$$

it implies that

$$\begin{array}{ll}\hfill {\delta }_n{∥u_n-x_n∥}^2& \le {∥x_n-z∥}^2-{∥y_n-z∥}^2-\left(1-{\delta }_n\right){∥v_n-x_n∥}^2\\ \le {∥x_n-z∥}^2-{∥y_n-z∥}^2\\ \le \left(∥x_n-z∥+∥y_n-z∥\right)∥x_n-y_n∥,\end{array}$$

by (3.12) and condition (i), we have

$$\underset{n\to \infty }{\text{lim}}∥u_n-x_n∥=0.$$

(3.19)

By using the same method as (3.19), we have

$$\underset{n\to \infty }{\text{lim}}∥v_n-x_n∥=0.$$

(3.20)

Since

$$\begin{array}{ll}\hfill {∥y_n-z∥}^2& \le {\alpha }_n{∥z_n-z∥}^2+\left(1-{\alpha }_n\right){∥z_n-z∥}^2\\ \le {\alpha }_n{∥x_n-z∥}^2+\left(1-{\alpha }_n\right){∥z_n-z∥}^2\\ \le {\alpha }_n{∥x_n-z∥}^2+\left(1-{\alpha }_n\right)\left({\delta }_n{∥M_n-z∥}^2\right.\\ \left.+\left(1-\delta \right){∥N_n-z∥}^2\right)\end{array}$$

(3.21)

Claim that

$$\underset{n\to \infty }{\text{lim}}∥A{u}_n-Az∥=\underset{n\to \infty }{\text{lim}}∥B{v}_n-Bz∥=0.$$

By nonexpansiveness of P _C , we have

$$\begin{array}{c}{\Vert y_n-z\Vert }^2\le {\Vert z_n-z\Vert }^2\\ \le {\delta }_n{\Vert P_C(I-\lambda A)u_n-P_C(I-\lambda A)z\Vert }^2\\ +(1-{\delta }_n){\Vert P_C(I-\eta B)v_n-P_C(I-\eta B)z\Vert }^2\\ \le {\delta }_n{\Vert (I-\lambda A)u_n-(I-\lambda A)z\Vert }^2+(1-{\delta }_n){\Vert (I-\eta B)v_n-(I-\eta B)z\Vert }^2\\ \le {\delta }_n{\Vert (u_n-\lambda A{u}_n-(z-\lambda Az)\Vert }^2+(1-{\delta }_n){\Vert (v_n-\eta B{v}_n-(z-\eta Bz)\Vert }^2\\ ={\delta }_n{\Vert (u_n-z)-\lambda (A{u}_n-Az)\Vert }^2+(1-{\delta }_n){\Vert (v_n-z)-\eta (B{v}_n-Bz)\Vert }^2\\ ={\delta }_n({\Vert u_n-z\Vert }^2+{\lambda }^2{\Vert A{u}_n-Az\Vert }^2-2\lambda 〈u_n-z,A{u}_n-Az〉)\\ +(1-{\delta }_n)({\Vert v_n-z\Vert }^2+{\eta }^2{\Vert B{v}_n-Bz\Vert }^2-2\eta 〈v_n-z,B{v}_n-Bz〉)\\ \le {\delta }_n({\Vert u_n-z\Vert }^2+{\lambda }^2{\Vert A{u}_n-Az\Vert }^2-2\lambda \alpha {\Vert A{u}_n-Az\Vert }^2)\\ +(1-{\delta }_n)({\Vert v_n-z\Vert }^2+{\eta }^2{\Vert B{v}_n-Bz\Vert }^2-2\eta \beta {\Vert B{v}_n-Bz\Vert }^2)\\ \le {\delta }_n({\Vert x_n-z\Vert }^2+\lambda (\lambda -2\alpha ){\Vert A{u}_n-Az\Vert }^2)\\ +(1-{\delta }_n)({\Vert x_n-z\Vert }^2+\eta (\eta -2\beta ){\Vert B{v}_n-Bz\Vert }^2)\\ ={\Vert x_n-z\Vert }^2-{\delta }_n\lambda (2\alpha -\lambda ){\Vert A{u}_n-Az\Vert }^2\\ -(1-{\delta }_n)\eta (2\beta -\eta ){\Vert B{v}_n-Bz\Vert }^2,\end{array}$$

it follows that

$$\begin{array}{ll}\hfill {\delta }_n\lambda \left(2\alpha -\lambda \right){∥A{u}_n-Az∥}^2& \le {∥x_n-z∥}^2-{∥y_n-z∥}^2\\ -\left(1-{\delta }_n\right)\eta \left(2\beta -\eta \right){∥B{v}_n-Bz∥}^2\\ \le \left(∥x_n-z∥+∥y_n-z∥\right)∥y_n-x_n∥,\end{array}$$

(3.22)

by conditions (i), (ii), λ ∈ (0, 2α) and (3.12), it implies that

$$\underset{n\to \infty }{\text{lim}}∥A{u}_n-Az∥=0.$$

(3.23)

By using the same method as (3.23), we have

$$\underset{n\to \infty }{\text{lim}}∥B{v}_n-Bz∥=0.$$

(3.24)

By nonexpansiveness of $T_{r_n}$, we have

$$\begin{array}{c}{\Vert M_n-z\Vert }^2={\Vert P_C(u_n-\lambda A{u}_n)-P_C(z-\lambda Az)\Vert }^2\\ \le 〈(u_n-\lambda A{u}_n)-(z-\lambda Az),M_n-z〉\\ =\frac{1}{2}({\Vert (u_n-\lambda A{u}_n)-(z-\lambda Az)\Vert }^2+{\Vert M_n-z\Vert }^2-\Vert (u_n-\lambda A{u}_n)\\ {-(z-\lambda Az)-(M_n-z)\Vert }^2)\\ \le \frac{1}{2}({\Vert u_n-z\Vert }^2+{\Vert M_n-z\Vert }^2-{\Vert (u_n-M_n)-\lambda (A{u}_n-Az)\Vert }^2)\\ =\frac{1}{2}({\Vert T_{r_n}{x}_n-T_{r_n}z\Vert }^2+{\Vert M_n-z\Vert }^2-{\Vert u_n-M_n\Vert }^2\\ +2\lambda 〈u_n-M_n,A{u}_n-Az〉-{\lambda }^2{\Vert A{u}_n-Az\Vert }^2)\\ \le \frac{1}{2}({\Vert x_n-z\Vert }^2+{\Vert M_n-z\Vert }^2-{\Vert u_n-M_n\Vert }^2+2\lambda 〈u_n-M_n,A{u}_n-Az〉\end{array}$$